Let $X,Y,Z$ be irreducible projective varieties over $\mathbb{C}$ (you can assume all of them are normal), and let $f:X\rightarrow Z, g: Y\rightarrow Z$ be two birational projective morphisms, such that

- there is a common open set $U\subset Z$ with $f_U:f^{-1}(U)\simeq U$ and $g_U:g^{-1}(U)\simeq U$,
- If $W:= Z-U$, then $f_W: f^{-1}(W)\rightarrow W$ and $g_W: g^{-1}(W)\rightarrow W$ are projective bundles of ranks $m,n$ respectively, and
- codim$(W)=1+m+n$.

Is it true that $X\times_{Z}Y$ is integral? I have seen a statement along this line, but can't prove it.

Any help would be really appreciated.

**Edit**: Thanks to R. van Dobben de Bruyn's comment, the answer to the question as I stated earlier was false. I had skipped the codimension condition.